A review and comparison of Bayesian and likelihood-based inferences in beta regression and zero-or-one-inflated beta regression

Abstract

Beta regression is an increasingly popular statistical technique in medical research for modeling of outcomes that assume values in (0, 1), such as proportions and patient reported outcomes. When outcomes take values in the intervals [0,1), (0,1], or [0,1], zero-or-one-inflated beta (zoib) regression can be used. We provide a thorough review on beta regression and zoib regression in the modeling, inferential, and computational aspects via the likelihood-based and Bayesian approaches. We demonstrate the statistical and practical importance of correctly modeling the inflation at zero/one rather than ad hoc replacing them with values close to zero/one via simulation studies; the latter approach can lead to biased estimates and invalid inferences. We show via simulation studies that the likelihood-based approach is computationally faster in general than MCMC algorithms used in the Bayesian inferences, but runs the risk of non-convergence, large biases, and sensitivity to starting values in the optimization algorithm especially with clustered/correlated data, data with sparse inflation at zero and one, and data that warrant regularization of the likelihood. The disadvantages of the regular likelihood-based approach make the Bayesian approach an attractive alternative in these cases. Software packages and tools for fitting beta and zoib regressions in both the likelihood-based and Bayesian frameworks are also reviewed.

Keywords

Penalized likelihood shrinkage prior zoib zero inflation one inflation correlated/clustered data

Get full access to this article

View all access options for this article.

References

Gheorghe

Brouwer

van Baal

. Did the health of the Dutch population improve between 2001 and 2008? investigating age- and gender-specific trends in quality of life. Eur J Health Econ 2015; 16: 801–811.

Kent

Gray

Schlackow

et al.

Mapping from the Parkinson's disease questionnaire PDQ-39 to the generic EuroQol EQ-5D-3L: the value of mixture models. Med Decis Making 2015; 35: 902–911.

Conrado

Denney

Chen

et al.

An updated Alzheimer's disease progression model: incorporating non-linearity, beta regression, and a third-level random effect in NONMEM. J Pharmacokinet Pharmacodyn 2014; 41: 581–598.

Tutoglu

Boyaci

Koca

et al.

Quality of life, depression, and sexual dysfunction in spouses of female patients with fibromyalgia. Rheymatol Int 2014; 34: 1079–1084.

Cordier

Goze

Schaedelin

et al.

A better surgical resectability of WHO grade II gliomas is independent of favorable molecular markers. J Neurooncol 2015; 121: 185–193.

Pammer

Kevan

. The contribution of visual sensitivity, phonological processing and non-verbal IQ to children's reading, Canberra: The Australian National University, 2004. [Unpublished manuscript.].

Schmid

Wickler

Maloney

et al.

Boosted beta regression. PLoS One 2013; 8: e61623.

Ospina

Cribari-Neto

Vasconcellos

. Improved point and interval estimation for a beta regression model. Comput Stat Data Anal 2006; 51: 960–981.

Simas

Barreto-Souza

Rocha

. Improved estimators for a general class of beta regression models. Comput Stat Data Anal 2010; 54: 348–366.

10.

Kosmidis

Firth

. A generic algorithm for reducing bias in parametric estimation. Electron J Stat 2010; 4: 1097–1112.

11.

Grün

Kosmidis

Zeileis

. Extended beta regression in R: Shaken, stirred, mixed, and partitioned. J Stat Software 2012; 48: 1–25.

12.

Ferrari

Cribari-Neto

. Beta regression for modelling rates and proportions. J Appl Stat 2004; 31: 799–815.

13.

Espinheiraa

Ferraria

Cribari-Neto

. On beta regression residuals. J Appl Stat 2008; 35: 407–419.

14.

Espinheiraa

Ferraria

Cribari-Neto

. Influence diagnostics in beta regression. Comput Stat Data Anal 2008; 52: 4417–4431.

15.

Smithson

Merkle

Verkuilen

. Beta regression finite mixture models of polarization and priming. J Edu Behav Stat 2011; 36: 804–831.

16.

Cribari-Neto

Zeileis

. Beta regression in R. J Stat Software 2010; 34: 1–24.

17.

Zeileis A, Cribari‐Neto F, Gruen B, et al. Beta regression. [R package version 3.0.5.], https://cran.r-project.org/web/packages/betareg/betareg.pdf.

18.

Swearingen C, Melguizo CM and Bursac Z. Modeling percentage outcomes: the % beta regression macro. SAS Global Forum 2011: Statistics and Data Analysis, paper 335.

19.

Cepeda-Cuervo E. Modelagem da variabilidade em modelos lineares generalizados. PhD Thesis, Instituto de Matem à tica. Universidade Federal do Rìo do Janeiro, Brazil, www.bdigital.unal.edu.co/9394/1/modelagemvariabilidademodelos.pdf. (accesssed 5 May 2016).

20.

Cepeda-Cuervo

Gamerman

. Bayesian methodology for modeling parameters in the two parameter exponential family. Estadistica 2005; 57: 93–105.

21.

Cepeda

. Beta regression models: Joint mean and variance modeling. J Stati Theor Pract 2015; 9: 134–145.

22.

Marin M, Rojas J and Jaimes D. Bayesian beta regression: joint mean and precision modeling. [R package version 1.2.], https://cran.r-project.org/web/packages/Bayesianbetareg/Bayesianbetareg.pdf (2014).

23.

Liu

Kong

. zoib: an r package for Bayesian inference for beta regression and zero/one inflated beta regression. R J 2015; 7: 34–51.

24.

Liu

. Bayesian model for joint analysis of multivariate repeated measures and time to event data in crossover trials. Stat Meth Med Res 2016; 25: 2180–2192.

25.

Liu

Kong

. Bayesian inference for zero/one inflated beta regression model. [R package version 1.3.3.], https://cran.r-project.org/web/packages/zoib/zoib.pdf (2015).

26.

Smithson

Verkuilen

. Better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychol Meth 2006; 11: 54–71.

27.

Cook

Kieschnick

McCullough

. Regression analysis of proportions in finance with self selection. J Empir Finance 2008; 15: 860–867.

28.

Kang

Delzell

DAP

Chhabra

et al.

Factors associated with high rates of antiretroviral medication adherence among youth living with perinatal HIV in Thailand. Int J STD AIDS 2015; 26: 534–541.

29.

Hatfield

Boye

Hackshaw

et al.

Models for survival times and longitudinal patient-reported outcomes with many zeros. J Am Stat Assoc 2012; 107: 875–885.

30.

Ospina

Ferrari

. A general class of zero-or-one inflated beta regression models. Comput Stat Data Anal 2012; 56: 1609–1623.

31.

Ospina

Ferrari

. Inflated beta distributions. Stat Papers 2010; 51: 111–126.

32.

Rigbyi

Stasinopoulos

. Generalized additive models for location scale and shape. Appl Stat 2005; 54: 507–554.

33.

Stasinopoulos

Rigbyi

. Generalized additive models for location scale and shape (gamlss) in r. J Stat Software 2007; 23: 1–46.

34.

Swearingen C, Melguizo CM and Bursac Z. Inflated beta regression: zero, one, and everything in between. SAS Global Forum 2012: Statistics and Data Analysis, paper 325.

35.

Stasinopoulos

Voudouris

BRV

. The library for fitting gamlss models. Comput Stat Data Anal 2015; 4: 3–4.

36.

Stata. Fractional outcome regression, www.stata.com/new-in-stata/fractional-outcome-models/(accessed 13 November 2015).

37.

Akaike H. Information theory and an extension of the maximum likelihood principle. In: Petrov BN and Csáki F (eds) 2nd international symposium on information theory, Tsahkadsor, Armenia, USSR, 2–8 September 1971, pp.267–281. Budapest: Akadémiai Kiadó.

38.

Hurvich

Tsai

C-L

. Regression and time series model selection in small samples. Biometrika 1989; 76: 297–307.

39.

Schwarz

. Estimating the dimension of a model. Ann Stat 1978; 6: 461–464.

40.

Spiegelhalter

Best

Carlin

et al.

Bayesian measures of model complexity and fit (with discussion). J R Stat Soc B 2002; 64: 583–639.

41.

Gupta

Saraless

. Handbook of beta distribution and its applications, New York, NY: CRC Press, 2004.

42.

Pinheiro

Bates

. Approximations to the log-likelihood function in the nonlinear mixed-effects model. J Comput Graph Stat 1995; 4: 12–35.

43.

Tibshirani

. Regression shrinkage and selection via the lasso. JRSS 1996; 58: 267–288.

44.

Hoerl

Kennard

. Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 1970; 12: 55–67.

45.

MacKay

Bayesian methods for backpropagation networks. In: Domany

van Hemmen

Schulten

(eds). Models of neural networks III: association, generalization, and representation, New York, NY: Springer-Verlag, 1996, pp. 211–254.

46.

Neal

. Bayesian learnings for neural networks, Canada: University of Toronto, 1994.

47.

Gelman

. Prior distributions for variance parameters in hierarchical models. Bayesian Anal 2006; 1: 515–533.

48.

Barnard

McCulloch

Meng

X-L

. Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage. Stat Sin 2000; 10: 1281–1311.

49.

O'Malleya

Zaslavskya

. Domain-level covariance analysis for multilevel survey data with structured nonresponse. J Am Stat Assoc 2008; 103: 1405–1418.

50.

Gelman

Rubin

. Inference from iterative simulation using multiple sequences. Statistical science. J Am Stat Assoc 1992; 7: 457–511.

51.

Brooks

Gelman

. General methods for monitoring convergence of iterative simulations. J Comput Graph Stat 1998; 7: 434–455.

52.

Neal

. Slice sampling. Ann Stat 2003; 31: 705–767.

53.

Plummer M. JAGS, http://mcmc-jags.sourceforge.net/ (accessed 5 May 2016).

54.

Lee

Song

. Evaluation of the Bayesian and maximum likelihood approaches in analyzing structural equation models with small sample sizes. Multivar Behav Res 2004; 39: 653–686.

55.

Maas

CJM

Hox

. Sufficient sample sizes for multilevel modeling. Methodol Eur J Res Meth Behav Social Sci 2005; 1: 86–92.

56.

Baldwin

Fellingham

. Bayesian Methods for the analysis of small sample multilevel data with a complex variance structure. Psychol Meth 2013; 18: 151–164.

57.

Cole

Chu

Greenland

. Maximum Likelihood, profile likelihood, and penalized likelihood: a primer. Am J Epidemiol 2014; 179: 252–260.

58.

Harville

. Maximum likelihood approaches to variance component estimation and to related problems. J Am Stat Assoc 1977; 72: 320–338.

59.

Laird

Ware

. Random-effects models for longitudinal data. Biometrics 1982; 38: 963–974.

60.

Laird

Lange

Stram

. Maximum Likelihood computations with repeated measures: application of the EM algorithm. J Am Stat Assoc 1987; 82: 97–105.

61.

McCulloch

. Maximum likelihood algorithms for generalized linear mixed models. J Am Stat Assoc 1997; 92: 162–170.